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Words and Polynomial Invariants of Finite Groups in Non-Commutative Variables
Anouk Bergeron-Brlek1, Christophe Hohlweg2, and Mike Zabrocki1
1Department of Mathematics and Statistics, York University, 4700 Keele Street, Toronto,
anoukbrlek@gmail.com, zabrocki@mathstat.yorku.ca
Colin.C.Adams@williams.edu
2LaCIM, Université du Québec à Montréal, CP 8888 Succ., Centre-ville, Montréal (Québec) H3C 3P8, Canada
hohlweg.christophe@uqam.ca
Annals of Combinatorics 16 (1) pp.1-36 March, 2012
AMS Subject Classification: 05C25, 05E10, 20C30, 20F55, 20F10
Abstract:
Let V be a complex vector space with basis {x1, x2, . . . , xn} and G be a finite subgroup of GL(V). The tensor algebra T(V) over the complex is isomorphic to the polynomials in the non-commutative variables x1, x2, . . . , xn with complex coefficients. We want to give a combinatorial interpretation for the decomposition of T(V) into simple G-modules. In particular, we want to study the graded space of invariants in T(V) with respect to the action of G. We give a general method for decomposing the space T(V) into simple modules in terms of words in a Cayley graph of the group G. To apply the method to a particular group, we require
a homomorphism from a subalgebra of the group algebra into the character algebra. In the case of G as the symmetric group, we give an example of this homomorphism from the descent algebra. When G is the dihedral group, we have a realization of the character algebra as a subalgebra of the group algebra. In those two cases, we have an interpretation for the graded dimensions and the number of free generators of the algebras of invariants in terms of those
words.
Keywords: non-commutative invariant theory, representation theory, finite groups, symmetric group, dihedral group, Cayley graphs, words

References:

1. Bergeron, N., Reutenauer, C., Rosas, M., Zabrocki, M.: Invariants and coinvariants of the symmetric groups in noncommuting variables. Canad. J. Math. 60(2), 266–296 (2008)

2. Chauve, C., Goupil, A.: Combinatorial operators for Kronecker powers of representations Sn. Sém. Lothar. Combin. 54, Art. B54j (2006)

3. Chevalley, C.: Invariants of finite groups generated by reflections. Amer. J. Math. 77, 778–782 (1955)

4. Comtet, L.: Advanced Combinatorics: The Art of Finite and Infinite Expansions. D. Reidel Publishing Co., Dordrecht (1974)

5. Dicks, W., Formanek, E.: Poincaré series and a problem of S. Montgomery. Linear and Multilinear Algebra 12(1), 21–30 (1982/83)

6. Humphreys, J.E.: Reflection Groups and Coxeter Groups. Cambridge Studies in Advanced Mathematics, 29. Cambridge University Press, Cambridge (1990)

7. Kharchenko, V.K.: Algebras of invariants of free algebras. Algebra and Logic 17(4), 478–487 (1978)

8. Lane, D.R.: Free Algebras of Rank Two and Their Automorphisms. Ph.D thesis, Bedford College, London (1976)]

9. MacMahon, P.A.: Combinatory Analysis. Vol. I, II. Cambridge University Press, Cambridge (1915/1916)

10. Molien, T.: Uber die Invarianten der linearen Substitutions gruppe. Sitz. Konig. Preuss. Akad. Wiss. 1152–1156 (1897)

11. Poirier, S., Reutenauer, C.: Algèbres de Hopf de tableaux. Ann. Sci. Math. Québec 19(1), 79–90 (1995)

12. Robinson, G. de B.: On the representations of the symmetric group. Amer. J. Math. 60(3), 745–760 (1938)

13. Rosas, M.H., Sagan, B.E.: Symmetric functions in noncommuting variables. Trans. Amer. Math. Soc. 358(1), 215–232 (2006)

14. Schensted, C.: Longest increasing and decreasing subsequences. Canad. J.Math. 13, 179–191 (1961)

15. Shephard, G.C., Todd, J.A.: Finite unitary reflection groups. Canad. J. Math. 6, 274–304 (1954)

16. Solomon, L.: A Mackey formula in the group ring of a Coxeter group. J. Algebra 41(2), 255–264 (1976)

17. Stanley, R.P.: Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (N.S.) 1(3), 475–511 (1979)

18. Wolf,M.C.: Symmetric functions of non-commutative elements. Duke Math. J. 2(4), 626–637 (1936)